To determine the constant of variation for a direct variation problem, we can use the relationship given by the direct variation equation:
[tex]\[ y = kx \][/tex]
Here, [tex]\( k \)[/tex] is the constant of variation. We start by solving for [tex]\( k \)[/tex] using the given point [tex]\((12, 9)\)[/tex], where [tex]\( x = 12 \)[/tex] and [tex]\( y = 9 \)[/tex].
First, we substitute the given values into the direct variation equation:
[tex]\[ 9 = k \cdot 12 \][/tex]
To isolate [tex]\( k \)[/tex], we divide both sides of the equation by 12:
[tex]\[ k = \frac{9}{12} \][/tex]
Now, simplify the fraction:
[tex]\[ k = \frac{3}{4} \][/tex]
Therefore, the constant of variation [tex]\( k \)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Given the answer choices:
- [tex]\(\frac{1}{2}\)[/tex]
- [tex]\(\frac{3}{4}\)[/tex]
- 1
- 2
The value of [tex]\(\frac{3}{4}\)[/tex] corresponds to the second choice.
So, the correct choice is:
[tex]\(\boxed{\frac{3}{4}}\)[/tex]
